Schedule-at-a-Glance

58 PRIMA 2013 Abstractsfunction globally constructed for the systems. In this talk,we first present a novel numerical constructive methodon the Lyapunov function for a wide series of dynamicalsystems. Then, we apply this method in some famousexamples, such as the Van Der Pol oscillator and Lorenzsystem. On the other hand, we analytically construct theLyapunov function in a chaotic system and the competitiveLotka-Volterra systems. The structure of the attractors,such as fractal, are demonstrated clearly by theLyapunov function.The half line versus finite domain problems ofthe Modified Buckley-Leverett equationYing WangUniversity of Minnesota, USAwang@umn.eduBuckley-Leverett (MBL) equation describes two-phaseflow in porous media. The MBL equation differs fromthe classical Buckley-Leverett (BL) equation by includinga balanced diffusive-dispersive combination. The dispersiveterm is a third order mixed derivatives term, whichmodels the dynamic effects in the pressure difference betweenthe two phases. The classical BL equation gives amonotone water saturation profile for any Riemann problem;on the contrast, when the dispersive parameter islarge enough, the MBL equation delivers non-monotonewater saturation profile for certain Riemann problemsas suggested by the experimental observations. In thistalk, we show that the solution of the finite interval [0, L]boundary value problem converges to that of the half-line[0, +∞) boundary value problem expoentially fast for theMBL equation as L → +∞. (This is a joint work withChiu-Yen Kao.)Blowup of classical solutions to the compressibleNavier-Stokes equationsWei YanInstitute of Applied Physics and Computational Mathematics,Chinawyanmath@gmail.comIn this talk, we present our recent results on the finitetime blow up of smooth solutions to the CompressibleNavier-Stokes system. We prove that any classical solutionsof viscous compressible fluids without heat conductionwill blow up in finite time, as long as the initial datahas an isolated mass group (see definition in the paper).The results hold regardless of either the size of the initialdata or the far fields being vacuum or not. This improvesthe blowup results of Xin (1998, CPAM) by removing thecrucial assumptions that the initial density has compactsupport and the smooth solution has finite total energy.Furthermore, the analysis here also yields that any classicalsolutions of viscous compressible fluids without heatconduction in bounded domains or periodic domains willblow up in finite time, if the initial data have an isolatedmass group satisfying some suitable conditions.Stability of supersonic contact discontinuitiesfor three dimensional compressible steady EulerflowsFang YuShanghai Jiao Tong University, Chinayufang0820@sjtu.edu.cnIn this talk, we discuss the nonlinear structural stabilityof supersonic contact discontinuities in three dimensionalcompressible isentropic steady Euler equations. Weobtain a necessary and sufficient condition for the linearstability of supersonic planar contact discontinuitiesand a priori estimates of solutions to the linearized problem.The weak stability of this contact discontinuityalso results in a loss of regularity with respect to thesource terms in the interior domain and on the boundaryin the a priori estimates of solutions to the linearizedproblem. Contact discontinuities with tangential velocityfields on two sides of the discontinuous front parallelor non-parallel are both considered. Moreover, using thecalculus of paradifferential operators and taking advantageof the control of non-characteristic components ofunknowns by the equations and the problems satisfied byvorticities of velocity fields, we get estimates of high orderderivatives of solutions to the linearized problem of a nonplanarcontact discontinuity. As there is a loss of regularityin the estimates of solutions to the linearized problem,we adapt the Nash-Moser-Hörmander iteration scheme toobtain the nonlinear stability of supersonic contact discontinuitiesin three dimensional compressible isentropicsteady Euler equations. This is a joint work with Ya-Guang Wang.Global structure of admissible BV solutionsto strictly hyperbolic conservation laws in onespace dimensionLei YuScuola Internazionale Superiore di Studi Avanzati, Italyyulei@sissa.itIn [1], we describe the qualitative structure of an admissibleBV solution to a strictly hyperbolic system of conservationlaws whose characteristic families are piecewisegenuinely nonlinear. More precisely, we prove that thereare a countable set of points Θ and a countable family ofLipschitz curves T such that outside T ∪Θ the solutionis continuous, and for all points in T \Θ the solution hasleft and right limit. This extends the corresponding structuralresult in [2] for genuinely nonlinear systems. Theproof is based on the introduction of subdiscontinuities ofa shock, whose behavior is qualitatively analogous to thediscontinuities of the solution to genuinely nonlinear systems.An application of this result is the stability of thewave structure of solution w.r.t. L 1 loc-convergence. Alsoa remark on the generalization of the results of generalstrictly hyperbolic conservation laws will be given.[1] S. Bianchini and L. Yu, Global structure of admissibleBV solutions to piece- wise genuinely nonlinear,strictly hyperbolic conservation laws in on space di- mension.To appear on Comm. Partial Differential Equations(arXiv:1211.3526), 2012.[2] A. Bressan and P. G. LeFloch, Structural stabilityand regularity of entropy solutions to hyperbolic systemsof conservation laws, Indiana Univ. Math. J., 48 (1999),pp. 43–84.Contributed Talks Group 6Probability and StatisticsEstimates of Itô integral and applications torandom dynamical systemsXiaoming FanSouthwest Jiaotong University and Beijing Normal University,Chinafanxm@swjtu.cnThis paper obtains two estimates of Itô integral and theirstochastic Gronwall’s inequalities, which present an ultimateway to deal with almost sure estimates for stochastic(partial) differential equations with white noises ofany type. As examples, they are applied to present a